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Literature review notes

% ==========================================================================================================================================

\section {Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination. J Geod (2009)}
\citep{jaggi1999}
The authors assess different methodologies for in-fight determination of empirical phase pattern corrections for LEO receiver antennas and discus their impact on Precise Orbit Determination (POD).

The authors show the use of empirically determined PCVs improve the quality of low degree Spherical Harmonic components of the earth's gravity field. Depending on the actual LEO receiver antenna pattern, a significant propagation of unmodelled PCVs  into the gravity field solution should be expected. It was shown that the antenna PCVs contribute significantly to the error budget of GRACE h1-SST tracking data. This might be of importance for the current GOCE mission where the low degree SH coeeficients are exclusively deterimined from GPS data.

High degree spherical co-efficients improvements are less pronounced. GPS signals are typically affected by similair systematic errors over a few minutes. Therefore errors of highest frequency are not introduced into kinematic POD when neglecting PCVs in the modelling of carrier phase obs.

PCO \& PCV values for the LEO antennas are obtained from ground calibrations and do not reflect the influence of error sources which are additionally encountered in the actual spacecraft environment ( near-field multipath, cross-talk with active GPS occulation antennas ).

Two empirical approaches were detailed in the paper, a residual and a direct approach for inflight determination. Simulations indicated that recovery of PCVs is difficult due to a comlex interaction between the recovery approach and the underling orbit parametization.

Simulations indicate that the residual approach is even more limited in the recovery of PCVs as GPS-specific parameters such as receiver clock error correction and carrier phase ambiguity partially absorb PCVs (particularly for large scale structures). Small scale structures could still be wel recovered with the residual approach.
Both techniques were limited as LEO positions have to be kept fixed on (degraded) trajectories to derive the PCVs iteratively

The author concludes that only the direct aproach has the potential to completely circumvent this problem by a simulatneous estimation of PCV, orbit and clock parameters as well as carrier phase ambguities. (BERN has to be upgrade it's physical models, etc).

% ==========================================================================================================================================

\section {Mapping the GPS multipath environment using the signal-to-noise raio (SNR)}
\citep{bilichSNRMap2007}

The authors develop a power spectral mapping methodology to visually represent the site specific multipath environment. They use the spectral content (frequency and magnitude) of SNR time series to deteremine which satellites and which portions of the antenna environment contribute significant multipath error and at what frequencies.

Diffuse Multipath, random reflections do not give a systematic bias. Specular reflection however will cause a systematic bias in GPS positioning results, in the order of several meters of pseudorange, and several cm for phase error.

The authors believe the SNR is a good observable to use for multipath determination as: \begin{itemize}
\item Easily reported in the RINEX format
\item Avoids positioning analysis, residual analysis
\item This avoids any intrinsic redistribution of mulitpath error
\item Avoids the need to Double Difference so can generate a site specific map
\item SNR is available from both frequencies, allowing an independent assessment of multipath effects on each frequency.
\end{itemize}
	
The authors point out shortcoming of other mulitpath determination techniqes:

TEQC MP1 \& MP2 \begin{itemize}
\item Single RMS is calculated for each site per day, no sense of direction or magnitude differences
\item Is more of a measure of pseudorange precision
	\begin{itemize}
	\item highly dependent upon proprietary smoothing
	\item does not relate to phase multipath
	\item changes or improvements are more likely due firmware than changes in multipath \citep{ray2005}
	\end{itemize}
\end{itemize}

Directional Antenna was able to precisely measure the orientation and magnitude of multipath effects, but this system has only been tested for L1 and requires reoccupation of the site if multipath conditions change. 

Ray Tracing Methodolgies are capable of depicting multipath directionality and magnitude when provided with precise measurements of all objects in the antenna environment, in reality this is rarely available in practice.

Multipath can be derived from time series analysis, but this is assuming this is purely repeating and correlated with the satellite geometry.

The authors give an overview of the Signal-to-Noise Ratio observable.
If persistent reflecting objects and all reflecting surfaces are horizontal change in SNR has the following relationship
dSNR/dt = 2Pi/lambda 2* h cos theta dtheata/dt

From the equation the following can be expected:
\begin{itemize}
\item Slow oscillations in SNR will result from nearby objects, 
\item whereas fast or high frequency oscillations in SNR are generated by distant objects.
\item Satellite motions substantial affects the eriod of multipath reflections ( cos theta dtheta/dt) 
\item Reflections from nearby objects may lead to a large range of expected multipath periods depending on the satellite
\item Multipath from distant reflections will results in a tighter range of possible periods
\end{itemize} 

The phase error will be maximized when a multipath phasor is at right angles to the direct signal (phi = 90 or 270). So the maximum possible phase error due to multipath is only a function of direct and multipath amplitudes:
dphimax = arctan(Am/Ad)

The authors use localized estimated of the multipath amplitudes and frequencies derived from a wavelet transform of the observed SNR values. By concentrating on a sectral band of interest the wavelet power at every epoch is assigned to a satellite location (azimuth and elevation), then combined with spectral data from all available satellites, this is then projected onto a sky plot.

Data Selection and Processing:\begin{itemize}
\item Convert the SNR values into Volts so that multipath signals are expressed in amplitude units
\item Assume the dominant trend is due to the direct signal, and multipath creates the smaller amplitude oscillations which are modulated intop of the Ad trend.
\item The best low order polynomial fit (order 3-15) is applied to the SNR time series and removed.
\item this leaves only the dSNR component which should be due to multipath
\item By assuming all long -period SNR contributions are due to the direct signal, the authors are concentrating on mapping contributions from objects with a larger h.
\item This may have the effect of removing multipath from near by objects.
\end{itemize}

\subsection{Future Work:}
\begin{itemize}
\item Filter the SNR time series before wavelet analysis to exclude frequencies of oscillation which are outside the range of periods sampled by the wavelet transform.
\item Polynomial removal scheme is non-optimal, frequencies of interest are being removed, and it may still be failling the zero-mean requirement for the wavelet transfrom.
\item When reflections from close and far objects occur at the same time the superposition of long and short period oscillations create difficulties fro the wavelet transform.
\item Filtering to optimize for a particular frequency would enable more accurate determination of amplitude and frequency\item Investigate the data quality issues of SNR such as quantization, and incorrect S1/S2 correlations.
\item Apply a SNO pattern from an antenna calibration to remove the satellite azimuth and elevation variation cause dby the direct signal. Compare an anechoic chamber derived SNO with a robot determined SNO ratio. Are there differences? Is this caused by the robot receiver choice?
\item Determine if a plot can be generated for a spectrum of distances after the frequency analysis
\item Investigate the use of different waveletes to optimze the filtering of the frequency band of interest. If we assume NF MP is slowly varying, but is realeted to sat elevation, then there may be an optimum wavelet to pick up this kind of non-stationary signal.
\item Look at using GNSS observations to densify the SNR plot, does this contribute to the technique significantly? Is there an effect from the different constellations? Is this due to an increase in observations, different frequencies or orbit paths (different range of geometries).
\end{itemize}

% ==========================================================================================================================================

\section{Geodetic Techniques for Time and Freqeuncy comparisons using GPS phase and code measurements}
\citep{rayGeodeticTechniques2005}
GPS carrier phase-based time transfer has a precision approaching 100ps over one day arcs. Absolute time transfer is limited to > 1ns due to instrument calibration uncertainties.
The quality of clock estimate is maximized by ensuring the longest possible spans of continuous phase data free of cycle slips (this minimises the number of ambiguities that need to be estimated).

Instrument and Hardware considerations
\begin{itemize}
\item configuration of receiver needs to be kept as simle as possible
\item aim for stability of system compnentsand their environment.
\item Changes should be made to a single component at a time, as this allows the opportunity assess the consequences
\item There are significant non-zero mean biases in the modulation of the GPS signal
\item the most important is the pseudorange bias P1 and P2. The peak-to-peak bias between P1 and P2 are more than 10ns.
\item The broadcast cloaks are determined of the ionosphere free P1/P2 linear combination, users must compensate for the P1-P2 bias by using the Tgd group delay bias given in the navigation message.IGS monitors and reports on DCBs coninuously on daily intervals.
\item Note that the TGD correction assumes P1-P2 bias is appropriate for single frequency users of C/A-code (the same as P1). This is not strictly true as there are P1-C/A bias have a peak-to-peak range of about 5ns. Other receivers report C/A + (P2 -P1) instead of true P2 which has a different bias.
\item Satellite transmit signals are assumed to come from an antenna to be perfectly hemispherical, there is storng evidence pointing to this assumption being significantly incorrect. This is thought to cause errors in the GPS frame scale (radial) at levels of approximately 10-15 ppb.
\end{itemize}

Site Selection
\begin{itemize}
\item The antena needs to be situated in such a way as to minimize reflections
\item The L2 signal is particular sensitive to reflections behind the antenna. If the antenna cannot be placed directly against a non-reflective surface, then it is usually best to put it as high above any background as practical.
\item the phase centre of the antenna, and hence the geodetic ref. point will depend on the direction of the signal from a particular satellite. Ngelecting these effects can cause systematic errors in station height determination of up to 10cm.
\item Radomes can distort the wavefron phases, whcih gives rise to an apparent shift in station position, especially height. Differences in position with and without the radome can reach levels of several cms. Conical radomes are particularly bad.
\item Antenna cables should be as short as possible and be as continuous as possible.
\item Inline amplifiers should be avoided
\item Need to ensure there is the best match for power and impedance
\item antenna cable should be burried and in conduit to minimze temperature effects.
\item The GPS receiver should be maintained in an environmentally controlled room.
\end{itemize}

Calibration
The internal delays within all the instrument hardware need to be accurately known to compare clock readings at one station to another.
\begin{itemize}
\item Absolute: an end-to-end set of bias measurements are made using a GPS simulator (which itslef must be calibrated)
\item Relative: differentially determine a side-by-side comparison against another simiair system taken as a stadnard reference.
\end{itemize}
The dominant source of errors source in the absolute calibration procedure is thought to be the GPS simulator itself. Subsequent differental calibrations against an absolute standard can be made at uncertainties of 1.6 ns, however the long term stabilit of these calibrated biases is not well known. Calibrated receivers can be adjusted to remove the instrumental bias in the process of generating the RINEX files.

Need to work out how much the group delay effect from the antenna may have an impact upon timing

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\section{Broadband Convex Impedance Ground Planes for Multi-System GNSS Reference Station Antennas}
\citep{tatarnikovTopcon2011}

The main purpose of a choke ring antenna is to provide protection against multipath reflected by the underlying terrain for ground based stations. However the choke ring ground plane contributes to some undesirable characteristics such as a narrowing of the of the antenna's directivity pattern in the elevation plane. This result sin a reduced tracking capability for sateliites at a low elevation. The choke ring performance is also frequency dependent. The groove depthshould slightly exceed a quarter of the carrier's wavelength. This is of particular importance due to theincrease in number of GNSS signals, and the larger bandwidth now required by Geodetic antennas.

The choke ring structure represents an impedance surface (this is the relationship between the electric and magnetic fields strenghts at the surface). For multipath rejection purposes the impedance of the ground plane surface is pure imaginary and capcitive. For traditional choke rings the impedance relationship is valid at the groove openings. The frequency response of the choke ring is determined to a large extent by the groove impedance frequency behaviour.

An alternative to the traditional choke ring is to use capacitive impedance, where the impedance surface is generated by a pin surface, or bed of nails. In this case the frequency devaitions of the impedance for the pin surface is much less compared to the choke ring.

the authors charcaterise multipath rejection capabilities , not in terms of CP signals, but in linear polarized signals. They asses the capability of their antenna using a Down / Up ratio (DU), this is the ratio between the antenna directivity pattern for angle theta below and above the antenna horizon, assuming a perfect electirc conductor reflecting surface.

DU (theta) = F ( - theta ) / F ( theta ) , this is expressed in dB units as DU(theta) (dB) = 20 log DU(theta). For antenna directivity patterns they use a transmitting mode rather than a receiving mode - equivalence is achieved as stated in the reciprocity theorem.

Impedance structure 

The groove in a choke ring can be viewed as a piece of coaxial waveguide shortened at the bottom end and open at the top.

According to the theory of waveguides, a coaxial waveguide could be characterized by a set of eigen waves (modes). 
Each mode has its eigen number. 
Modes with
\[ \phi_m < 2 \pi / \lambda \] 
can propogate, modes with \[\frac{phi_m > 2 * \pi}{\lambda} \] are evanescent. 
The first groove with radius R1 largely defines the ground plane frequency behaviour.

Another type of impedance structure, the pin structure, potentially allows for more homogenous and consistent frequency response wihtin the GNSS band as opposed to the common groove structure of the choke ring.

Maths gets very technical after this point in the paper!

Convex impedance ground plane allows for an antenna directivity pattern widening plane without signfiicantly affecting the calculate and measured multipath rejection capabilities.

% ==========================================================================================================================================

\section{GPS Antenna Design and Performance Adavancements: The Trimble Zephyr}
\citep{krantzTrimbleZephyr2001}

The Manufacturers claim that the antenna phase center repeatability and multipath resistance is comparable to the choke ring antenna, with improved tracking at low elevation.

The underlying design of the antenna is to be independent of frequency.

The effective electrical center of the antenna moves around in three dimensions. This is a function of the apparent azimuth and elevation of the satellite being tracked, and the way the antenna i sbeing fed, that is how it is electrically connected to the rest of the circuit.

In high performance antennas the variation of th ephase center in the horizontal plane islimited to less than 1mm regardless of the signal's direction. In low grade antennas the charcateristics of the antenna are not consistent over the asymmetrical horizontal surface of the antenna, so it wille xperience phase shifts that cary depending on the diretion of the satellite and antenna. Furthermore the low grade antennas typically do not have a strict manufacturing process, and 2 antennas from the same manufacturer may vary significantly in performance ans behaviour.

Antenna models which have a good repeatability will have almost identical phase center variations. In DD porcessing this will be highly correlated, and the error wil cancel out over short baseline lengths. As the baseline lenght increases, greater than 50km, the apparent satellite elevation and azimuth gradually become different for the antenna at each end of the line, that is the apparent elevation angle de-correlates with distance.

If an antenna phase centre moves very predicatbly as a finction of satellite elevation, regardless of the azimuth of the satellite, and if this consistent behaviour is held within very fine tolerances across all antennas manufactured, then this antenna has a very good phase center repeatability.

Patch antennas require different dimensions fo reach patch to be sensitive at different frequencies. They are often stacked ontop of each other, while each patch is connected to the antenna LNA usually by a single feed point. RHCP is achieved in the conventional single feed point by creating a 90 phse shift in two resonant modes on the patch. With a square patch antenna this can be achieved by making two parallel sides resonant under the center frequency, and two parallel sides resonant just over the center frequency. The drawback is that this will only work well for signals being tracked at the center frequency.
As the patch is non-symmetrical, it loses the CP required to effectively track RHCP of GPS signals. This loss of polarization leads to a reduced SNR. An antenna which is better tuned to RHCP, is more likely to reject LHCP and therefore more likely to reject MP.

The Zephyr antenna uses several feed points to maintain CP.

Ground plane design
RF signals can reflect off water, etc, the MP signal distorts the correlation peak from which the pseudorange measurements are made, a Ground Plane can help to reduce these effects. A flat metal ground plane is effective than no ground plane, but is has an inherant weakness in that it is a good conductor with low electrical resistance. If MP signals reflected from the ground strike the underside of the ground plane, at certain angles the ground plane's electrical properites can actually conduct multipath to the antenna through diffratcion or reflection.

The choke ring design consists of deep concentric wells in the ground plane, typically of a depth of 1/4 of a wavelength of which the signal to the antenna is being tuned. These wells acts as traps for signalsreflected from objects near the ground. The drwaback is the required depth of the CR is a function of frequency, so the antenna usually has to compromise between two frequencies reducing the effectiveness for both. This will become even more of a problem as more GNSS frequencies become available to track. CR will also attenuate Direct Line of site signals from low elevation satellites along with the offending MP, reduncing the SNR at low elevation.

The stealth ground plane improves the conventional flat metal ground plane, and is frequency independent. It relies on electrical resistnace to weaken the multipath signals. Anintricate pattern of tiny concentric rings exponentially increases electricla resistance as the distance from the center of the antenna increase. This simulates a ground plane of inifinite size.

The stated advantages were stated as 
\begin{itemize}
\item The design is not frequency dependent
\item Low elevation tracking is not compromised
\item The antenna is compact and light weight
\item MP resistant
\end{itemize}

The author's rely on 'unrigorous' testing techniques, no spectral analysis of the data for MP, reliance on TEQC as a definitive answer (ie avrg MP1 and average MP2), etc

% ==========================================================================================================================================
\section{A Novel GPS Survey Antenna}
\citep{waldemarPinWheel2000}
 The author decribes the antenna as a 'fixed beam phased array of aperture coupled slots optimized to receive a right hand polarized signal'. 
The antenna is made from:
\begin{itemize}
\item mode from a single PCB board
\item another PCB board is placed underneath the antenna to act as a reflector to reinforce the antenna directivity and reduce back lobe radiation.
\item radiation pattern roll-off is sharper than conventional patch antennas mounted in a choke ring (no comparison made with a di-pole in a choke ring)
\item claims no alignment is required with respect to true north due to the symmetry of the radiation pattern.
\item there is no phase offset between L1 and L2 frequencies
\item due to it's planar structure it can be buried into the body of a vehicle.
\end{itemize}

Ideally the antenna should receive only signals above the horizon and reject signals below the horizon plane of the antenna, have a known and stable phase center that is co=located with the geometrical centre of the antenna, and have perfect circular polarization charcteristics to maximise the reception of the incoming RHCP signal.

Axial Ratio (AR) is a typical measure of antenna polarization characteristics.

For multiple frequencies the antenna should have a common phase center at both frequencies, and ideally the same radiation pattern and axial ratio charcteristics.

Compares pin wheel to a Choke Ring Ground Plane Antenna (using a patch element) using the following metrics:

\begin{itemize}
\item Axial Ratio
\item Antenna amplitude directivity pattern roll-off (elevation plane)
\item Antenna amplitude directivity pattern varation in azimuth plane
\item Phase center location
\item Phase center variation
\end{itemize}

Choke ring ground planes
Gives a very high impedance plane that does not support image currents generated within the ground plane that would normally interfere with the currents generated within the patch antenna itself. this translates into very sidelobes underneath the antenna horizon and very smooth amplitude and phase patterns generated by the antenna, In addition a very good axial ratio value ( < 3 dB) above 10 elevation are achieved. The large size of the ground plane on the other hand, gives a sharper amplitude roll-off (from zenith to the horizon) and an increase main beam directivity.

A typical roll off is 10 to 12 dBi from zenith to horizon as compared to 3-5 dB roll-off of the patch antenna.

The phase center offset is inherited from the patch antenna itself, they use a stack configuration in order to resonate at 2 frequencies.
Microstrip patch antennas with stable phase centers must be fed in at least 2 points, preferable in four points (for all four edges of rectangular/square pacth). Since the antenna must be circular polarized, a 90 phase gradient must be established between the feedin points, as the number of feed pints increase so does the field network increase in complexity and lossy.

The pinwheel antenna is not a patch antenna but an array of multiple spiral slots that are electromagnetically coupled to a feedin network, the antenna uses 12 spirals for a dual-frequency version, 6 spiral arams resonate at L1, 6 spiral arms resonate at L2. The L1 and L2 spiral arms are interleaved to maintain a natrual symmetry of the antenna in all three axes. The antenna is made out of a flat printed circuit board (PCB) with the upper layer being the ground plane.

Having the ground plane as the top layer provides an additional shielding against electromagnetic pulse generate dby lightning, etc. In addition the electric field is forced to 0 (boundary condition) for the signals tangential to the ground plane (antenna horizon) . This gives a sharp amplitude pattern roll-off near the horizon, available only with choke ring antennass, or extremely large ground planes (18-20").

The typical amplitude roll off is 12 dB at L1, 15 dB at L2. Very light 500g and a small diameter of 6.25".

There is no phase offset between L1 and L2 bands for all three axes due to:
\begin{itemize}
\item Natural symmetry of all slots wrt to the geometrical center of the antenna
\item L1 and L2 slots are interleaved to maintain symmetry between bands
\item All L1 and L2 slots are contained in the same z axis plane
\end{itemize}


To maintain a good front/back ratio a thin metal reflector is placed underneath the antenna board. This prevents MP generated replicas be amplified by the antenna. 

The antenna phase center does move in a vertical plane for each channel which is inherently unavoidable with any antenna beam that is highly directive. A perfect phase center antenna of an imaginary ideal antenna would have to have the same amplitude and phase pattern in every direction. This would cause the antenna to be severly susceptible to multipath. So a compromise must be made to shape the amplitude radiation pattern to reduce then reception of MP at low elevations.

% ==========================================================================================================================================
\section{Innovation: GNSS Antennas An Introduction to Bandwidth, Gain Pattern, Polarization, etc}
\citep{moernautInnovationAntennas2009}

An antenna must be designed for the particual signals to be intercepted with the center frequency, bandwidth and polarization of the signals being important parametrs in the design process.

An antennas job is to capture some of the power in the electromagnetic wave it receives and to convert it into an electrical current that can be processed by the receiver. With increasin levels of precision, the increase in the performance required from the antenna to deal with multipath suppresion and phase-center stability becoming important charcateristics. It is true some of the antenna gain issues can be overcome by using the receiver's processing power and knowledge of the pseudo random noise spreading codes used to transmit the signals.

A number of important antenna properties
\begin{itemize}
\item Frequency Coverage
\item Gain Pattern
\item Circular Polarization
\item Multipath Suppression
\item Phase Center
\item Impact on receiver sensitivity
\item Interference Handling
\end{itemize}

Frequency Coverage:
GNSS receiver may now require GPS L5, Galileo E5/E6 and GLONASS bands. As the bandwidth requirement of an antenna increases, so does the complexity of the antenna design. This becomes even harder if a small antenna diameter is required.

Gain Pattern:
Gain is the ratio of the radiation intensity in a given direction to the radiation that wouild be obtained if the power acepted by the antenna was radiated isotropically. Variation of an antenna's gain is referred to as the radiation pattern or the receiving pattern. Under the reciprocity theorem these patterns are identical for a given antenna. 
The receiver operates best with only a small difference in power between the signals from the various satellites being tracked and ideally the antenna covers the entire hemisphere above it with no variation in gain. This has to do with potential cross-correlation problems in the receiver and the simple fact that excessive gain roll-off may cause signals from satellites at low elevation to drop below the noise floor of the receiver. On the othe rhand, optimization for multipath rejection and antenna noise temperature require some gain roll off.

Circular Polarization: 
Spaceborne receivers typically use circular polarization (CP) signals for transmitting and receiving. The changing relative orientation of the transmitting and receiving CP antennas as the satellites orbit the Earth will not experience polarization fading (as would be the case for linear polarized signals/antennas). IN addition CP does not suffer from the effects of Faraday rotation caused by the ionosphere. Faraday rotation results in an electromagnetic wave from space arriving at Earth's surface with a different polarization angle than it would have if the ionopshere was absent. This will leading to signal fading and potentially poor receprion of linearly polarized signals.

CP signals may either be right-handed or left handed, GNSS satellites use RHCP. Generally even if an antenna has been optimized for RHCP signals, they will still pick up some LHCP energy (also refered to as the cross-polar component)

The quality of the CP can be described by either specifying the ratio of this cross-polar component with respect to the co-polar component, or by specifying the axial ratio (AR).
AR is the measure of the poarization ellipcity of an antenna designed to receive CP signals, an AR close to 1 (or 0 dB) is desirable . For geodetic antennas the typical AR at zenith should not be greater than 1 dB, normally AR will increase towards lower elevation angles, for a geodetic antenna we would expect less than 3 to 6 dB. There mat be some small (< 1 dB) variations of AR versus azimuth at low elevations. Maintaining a good AR over the entire hemisphere and at all frequencies requires a lot of surface area.

Multipath suppresion:
An MP signal can come from 3 basic directions:
\begin{enumerate}
\item The gorund and arrive back at the antenna
\item The ground or an object and arrive at the antenna at a low elevation angle
\item An object and arrive at the antenna at a high elevation angle
\end{enumerate}

Reflected signals typically contain a large LHCP component

For ground reflections:

\[ 
MPR = \frac{ E_RHCP(\theta) } { E_RHCP(180-\theta) + E_LHCP(180-\theta)}
\]

The MPR for signals that are reflected from the ground surface equals the RHCP antenna gain at angle theta divided by the sum of RHCP and LHCP antenna gains at the supplement of the angle. 

Signals that are reflected from the ground require the antenna to have a good front-to-back ratio if we want to suppress then because a RHCP antenna has by nature a LHCP response in the back hemisphere.

For Vertical Reflections:
\[
MPR = \frac{E_RHCP (\theta)}{ E_RHCP(\theta) + E_LHCP(\theta)}
\]
The MPR for signals from reflections against vertical objects can be suppressed by having a good AR at those elevation angles. In this case the MPR simply equals the co-tocross-polar ratoi. LHCP reflections that arrive at high elevations are not a big problem because the AR tends to be good at these angles, and the reflections will be suppressed.

LHCP signals that arrive at lower elevations willpose more of a problems as the AR of the antenna is more likely to have been degraded. 
It makes sense to have some level of gain roll-off towards the lower elevation angles to help suppress multipath signals. However a good AR is a must as gain-roll-off alone will not do it.

Phase Center: 
The phase center is the point in space where all the rays appear to emanate from (or converge on) or it is the point where the electromagnetic fileds from all incident rays appear to add up in phase.
Ideally the phase center is a single point in space for all directions at all frequencies. However antennas will often possess multiple phase center point (for each lobe in the gain pattern) or a phase center that appears smoothed out as frequency and viewing angle are varied.

The PCO can be represented in 3D where the offset is specified for every direction at each frequency band.
For high end antennas phase center variations in azimuth are small and on the order of a couple of mm. The vertical offsets are typically 10mm or less. 
Best to calibrate and attempt to remove this affect.

Impact of Receiver Sensitivity:
The signal strength from space is typically around -130 dBm
A high performance low noise amplifier (LNA) between the antenna element and the receiver.
Can characterize the performance of a particular receiver element by its noise figure (NF), which is the ratio of actual output noise of the element to that which would remain if the element itself did not introduce noise.

The total (cascaded) noise figure of a receiver system (a chain of elements or stages) can be calculated using the Friss formula:

\[
NF = NF_1 + \frac{NF_2 -1}{ G_1} + \frac{NF_3 -1} {G_1* G_2} + ...
\]
The total system NF equals the sum of the NF of the first stage \[(NF_1)\] plus that of the second stage \[(NF_2)\] minus 1 divided by the total gain of the previous stage \[(G_1)\] and so on. So the total NF of the whole system equals that of the first stage plus any losses ahead of it such as those due to filters, etc.
We would expect to see LNA noise figures in the 3 dB range for high performance antennas.

The other requirement for the LNA is to have sufficient gain to minimize th eimpact of long and lossy coaxial antenna cable , typically 30dB should be enough.

It is important to have a balance gain as:
\begin{itemize}
\item too much gain may overload the receiver and drive it into non-linear behaviour (compression) degrading it's performance
\item too little and low elevation angle observations will be missed.
\end{itemize}

Interference Handling:
Careful desing of the antennacan help prevent interference by introducing some frequency selectivity against out-of-band interefers. The mechanisms by which inband and out-of-band interference can create trouble in the LNA and the receiver, and the approach to dealing with then are somewhat different.

Out-of-band interference:
Is generally from an RF source outside the GNSS frequency band (mobile base station, mobile phones, broadcast transmitters, radar, etc.
When these signals hit the LNA they can drive the amplifier into its non-linear range and the LNA starts to operate as a mulitplier or comb generator.

Through a similair mechanism 3rd order mixing porducts can be generated whereby a signal is multiplied by two and mixed with another signal. 
For instance take an airport radar operating at 1275 and 1305 MHz. Both signals double to 2550 ans 2610 MHz, these will in turn mix with the fundamentals and generate 1245 and 1335 MHz signals.

Another mechanisms is de-sensing, as the interference is amplified further down the LNA stages, it's amplitude increases and at some point the GNSS signals get attenuated because the LNA goes into compression. The same things may happen down the receiver chain. This effectively reduces the receivers sensitivity and in some cases reception will be lost completely.

RF filters can reduce out-of-band signals by 10s of dBs and this is sufficient in most cases. However the introduction of filters add insertion loss, amplitude and phase ripple, all of which degrade receiver performance.

In band Interference:
can be 3rd order mixing or simply an RF source that transmits inside the GNSS bands. If these interfers are weak the receiver can handle it, but from a certain power level on, there is not a lot that can be done.

The LNA should be designed for a high interception point (IP) at which non-linear behaviour begins (to prevent compression), there is now requirement for the LNA to be a power amplifier. 
LNAs with a higher IP tend to consume more power, so may not be practical for a rover antenna, but low current consumption should not be a requirement for a base station.


% ==========================================================================================================================================
\section{GPS Antenna LNA}
\citep{orbanGPSLNA2005}

Terminology:
Noise Figure (NF) - is a ratio that indicates how much noise power the LNA will contirbute to the total receiver noise. 

The minimum discemable signal (MDS) is the weakest signal a receiver can decode.

The more noise the LNA contributes, the higher the noise floor and the less sensitive the receiver is.
A poor LNA degrades the MDS of the receiver.
While the LNA NF is not the only factor that drives MDS, it is an important factor, since it is the single largest contributor to the system nois figure.
A typical state-of-the-art, single stage commercial grade LNA at L-Band (GPS) has a noise figure between 0.5 and 1 dB
A multistage GPS LNA with filtering has a noise figure between 1.0 and 2.5dB

Gain - this is the ratio of the input power to output power (slightly different to antenna definintion)
A single stage LNA is typically 15dB
Typical GPS LNAs use two or three gain blocks which yield between 25 to 50 dB of gain.
Unlike NF a low or high gain does not indicate a good or bad LNA.
It is important to specify the amount of gain that is required rather than go for the highest amount.
Too much gain will produce more intermodulation products in the LNA
Not enough gain may cause the GPS signal to be below the MDS level of the GPS receiver.

Second or Third Order Interception Point and 1dB Compression Point

These terms define the LNA's behaviour when either multiple (possible strong) signals or at least one stron signal is presented at it's input terminal.
Multiple signals in an LNA can mix and generate a set of new signals, some of which may lie in the GPS passband and cause interference.
A well designed LNA should be able to keep going with as much as -15 dBm of power at its terminal.

Supply Voltage
GPS receivers will put out a voltage to the RF input port to feed the LNA.
The amplifiers inside the LNA typically operate at 3 to 5V, which a voltage regulator in the LNA will convert from 12V/24 V
Need to be careful the LNA can handle the voltage being supplied.

Power Consumption
The current flowing through the LNA multiplied by the voltage being supplied.
For a single stage LNA the current is between 15mA to 100mA
A low current device means a poor intercept behaviour of the LNA

Filtering
Connecting an antenna to a GPS receiver using an LNA without any filtering in between in not a good idea. If reliability of reception in important.
All signals in a wide frequency range will enter the amplifier and generate spurious signals and degrade reception.
In addition strong out of band signals will degrade the sensitivity of the system by driving the LNA and possibly the receiver beyond their 1 dB compression point.
A wide range of filter options exist, SAW and ceramic are typical choices.
For L1 and L2 antenna normally use a duplexer to filter both boards.

There are several ways to implment filtering, depending on what is wanted to be achieved, here are some examples:

Example 1 - LNA with no filtering
\begin{itemize}
\item gives the best possible NF and gain
\item cheapest LNA
\item no out-of-band rejection
	\begin{itemize}
	\item any strong interference will cause compression and intermodulation
	\item multiple weaker signals will probably have the same effect.
	\end{itemize}
\item even though the second stage LNA has a higher noise figure, the system is still pretty close to that of the first stage
	\begin{itemize}
	\item This is why you can install a lossy coax cable after an LAN and not kill your system NF.
	\end{itemize}
\end{itemize}


\begin{tabular} { c | c | c | c }
         & LNA 1 & LNA 2 & Total\\
NF(dB) & 0.50 & 1.50 & 0.54\\
Gain(dB) & 16.00 & 15.0 & 31.00\\
OiP3(dBm) & 36.00 & 39.0 & 38.73\\
OP1 (dB) & 20.00 & 18.0 & 18.0\\
Input Pwr (dBm) & & & -95.00\\
System Temp (K) & & & 290.00\\
\end{tabular}

Example 2 - Upfront filter  
\begin{itemize}
\item has a filter with a lot of rejection upfront
\item if you are operating the LNA near a radar station, then this is a good choice
\item the NF suffers but the system is robust
\item possible to use a mechanical filter (cavity or similar) to minimize insertion loss
\item will be costly and heavy
\end{itemize}

\begin{tabular} { c | c | c | c | c}
         & Filter 1 & LNA 1 & LNA 2 & Total\\
NF(dB)   & 2.00 & 0.50 & 1.50 & 2.54 \\
Gain(dB) & -2.00 & 16.00 & 15.00 & 29.00\\
OiP3(dBm) & & 36.00 & 39.00 & 38.73\\
OP1 (dB) & & 20.00 & 18.00 & 18.00\\
Input Pwr (dBm) & & & & 95.00\\
System Temp (K) & & & & 290.00\\
\end{tabular}

Example 3 - filtering distributed across the LNA
\begin{itemize}
\item a good compromise between good rejection and best possible NF
\item NF is degraded with the insertion loss of the filter ahead of the LNA
\item This can be minimised by selecting a filter with less loss and less rejection
\item second filter adds rejection while the NF stays the same
\end{itemize}

\begin{tabular} { c | c | c | c | c | c }
         & Filter 1 & LNA 1 & Filter 2 & LNA 2 & Total\\
NF(dB)   & 0.50 & 0.50 & 0.50 & 1.50 & 1.06\\
Gain(dB) & -0.50 & 16.00 & -0.50 & 15.00 & 30.00\\
OiP3(dBm) & & 36.00 & & 39.00 & 38.70\\
OP1 (dB) & & 20.00 & & 18.00 & 18.00\\
Input Pwr (dBm) & & & & & -95.00\\
System Temp (K) & & & & & 290.00\\
\end{tabular}

Example 4 - variation of example 2
\begin{itemize}
\item still a lot of filtering in a single filter but placed after the first gain block
\item design would be to pick a very robust (high IP and high current) first stage so it can handle strong signals at input
\end{itemize}

\begin{tabular} { c | c | c | c | c}
         & LNA 1 & Filter 1 & LNA 2 & Total\\
NF(dB)   & 0.50  & 2.00  & 1.50 & 0.62\\
Gain(dB) & 16.00 & -2.00 & 15.00 & 29.00\\
OiP3(dBm) & 36.00& & 39.00 & 38.59\\
OP1 (dB) & 20.00 & & 18.00 & 18.00\\
Input Pwr (dBm) & & & & -95.00\\
System Temp (K) & & & & 290.00\\
\end{tabular}

Example 5 - variation of example 3

\begin{itemize}
\item filtering is distributed across the LNA, but noe ahead of the LNA
\item This behaves like example 4 
\end{itemize}

\begin{tabular} { c | c | c | c | c | c }
         & LNA 1 & Filter 1 & LNA2 2 & Filter 2 & Total\\
NF(dB)   & 0.50 & 0.50 & 1.50 & 0.50 & 0.56 \\
Gain(dB) & 16.00 & -0.50 & 15.00 & -0.50 & 30.00\\
OiP3(dBm) & 36.00 &      & 39.00 &       & 38.20\\
OP1 (dB) & 20.00  &      & 18.00 &       & 17.50\\
Input Pwr (dBm) & & & & & -95.00\\
System Temp (K) & & & & & 290.00\\
\end{tabular}

Note there will be variations in the NF and gain at different frequencies.


Overview of examples:

\begin{tabular} { c | c | c | c | c | c | c }
      & Total NF & Total Gain & OiP3 & OP1 & Input Power & System Temp(K)\\
Example 1 & 0.54 & 31.00 & 38.73 & 18.0 & -95.00 & 290.00\\
Example 2 & 2.54 & 29.00 & 38.73 & 18.0 & 95.00  & 290.00\\
Example 3 & 1.06 & 30.00 & 38.70 & 18.0 & -95.00 & 290.00\\
Example 4 & 0.63 & 29.00 & 38.59 & 18.0 & -95.00 & 290.00\\
Example 5 & 0.56 & 30.00 & 38.20 & 17.50 & -95.00 & 290.00\\
\end{tabular}

\begin{itemize}
\item if you need high gain make sure you add some filtering, preferably ahead of the first stage.
\item if you need filtering and a low noise figure distribute the filtering across the LNA stages.
\item if power consumption is not an issue, go for the highest Interception point
\item if you are using machinery guidancedo not compromise on the strong handling capabilities of the LNA
\item if you opt for lots of gain make sure your receiver can handle it.
\item ensure the receiver can output the right amount of current required by the LNA. 
\end{itemize}

Questions to follow up?
\begin{enumerate}
\item How does the LNA behave when there is a lack of current/ increase in resitance?
\item Put together a table of antenna and receiver values wrt to LNA metrics
\begin{itemize} 
\item max gain 
\item minimum discemable signal (MDS) 
\item type of LNA in each antenna, etc
\end{itemize}
\item what are the frequency response issues for GNSS? ie L5 Galileo
\item What does the INput Power mean in the tables?
\item What does the System Temp mean in the tables?
\item Need to better define 1 dB and INterception point nto really clear from this paper
\end{enumerate}
% ==========================================================================================================================================
\section{The importance of correct antenna calibration models for the EUREF Permanent Network}
\citep{volksenEurefAntenna2006}

The correction models for relative and absolute antenna calibrations differ significantly.
The introduction of the new 'absolute' antenna models is only possible in connection with new calibration models for the GPS satellite antennas.
This will stop an artificical scale factor being introduced.
The correction model with the radome is important for processing otherwise systematic effects are introduced mostly in the height components.
The dominant antenna type used in the IGS network is the AOAD.
This antenna served as a reference for the calibration of other antenna types in relative mode.
For the AOAD onlu offsets in the height component were assumed for each frequency, and no directional PCV was applied.
A database of relative calibrated antenna types was generated by Mader \citep(maderRelativeCal1999) which was regularly updated.

The main issue in th euse of relative calibrations is that the corrections are dependent on a reference antenna.
The estimation of PCV at low elevation angles is not possible due to the appearance of much more noise and stronger MP effects. 
This produces unwanted signals in the observation data set.
The environment used for the antenna calibration is never completely free of MP.
Satellite constellation at the location of calibration might not cover the entire hemisphere evenly. 
This can be improved by rotating the antenna, but MP will still remain.

Schuper and Clark \citep{schulperAnechoic1994} developed a calibration procedure in an anechoic chamber for PCO and PCV estimation.
But the method was complicated and not applicable for larger sets of antennas. (or is this really a resources issue, lack of access to a chamber?)

In 1996 \citep{wubbenaAbsCal1996} a new approach for the absolute calibration of GPS antennas was presented by the University of Hannover and the company Geo++.
The technique was developed further into an automatic real-time calibration of GPS antennas using a robot \citep{wubbenAbsCal2000}.
The technique allows for the estimation of the PCV in a MP-free environment.
It can also estimate the PCV down to an elevation of 0 degrees.
Estimates for azimuth and elevation dependent PCV.
Type specific corrections are availabel from a database, but not all results are freely available.

The calibration of GPS antennas in anechoic chamber was also revied  by Gorres \citep{gorresAnechoic2006}. They estimated independently the absolute PCV of GPS antennas in an anechoic chamber. 
The results proved to be consistent with the robot calibration.

There is a scale factor between coordinate sets compiled with relative models and absolute models. 
Schmid \citep{schmidSatAnt2003} presented a solution for the scale problem.
They estimated elevation dependent satellite antenna phase center variations based on absolute PCV for the GPS ground Antennas.
Applying these corrections for the GPS satellite antennas compensates the scale problem.

The use of absolute calibration models will have an impact on the EUREF reference system.
Not only will the systematic error caused by the conversion to absolute PCV, but also the introduction of correction models for antennas with domes will leave a significant trace.

Computations:
\begin{itemize}
\item Used the test igs05 file.
\item This had 10 absolute calibrated antennas by the robot GEO++.
\item 12 antennas calibrated by NGS converted to Abs PCV.
\item Sat Antenna Z-offsets (sat -specific)
\item Sat Patterns (block specific) determined for re-analysis of 11 years of data.
\item Scale of reference Frame was fixed to the network solution to IGB00.
\end{itemize}

In order to estimate the impact of the conversion from relative to absolute calibrated antennas two sets of computations were carried out.
\begin{itemize}
\item Coordinates were estimated for the BEK sub-network of EUREF
\item This consists of 70 stations
\item Relative calibrations were used fo rthe antennas taken from igs\_01.pcv
\item data from GPS week 1317 was processed using Bernese 5.0
\item daily solutions were computed applying a minimum constraint condition for the translation and the rotation of the network based on MATE, NICO, VILL, WTZR
\item Same processing was redone, except absolute antenna corrections were used in place of relative calibrations.
\end{itemize}

The impact of the Radomes is fairly large in the height component.
Identical antenna types show similari levels of displacement.

To understand only the impact of the conversion from relative to absolute PCV one should look at those sites where an AOAD antenna is installed without a DOME.
The horizontal displacements are negligible.
Vertical displacements are apparent but not significant compared to other antenna vertical displacements.

Looking at the average displacement by antenna type, you will see that the standard deviation is larger compared to the displacement signal.
This indicates you cannot just apply a correction values to the coordinates for each antenna type as there are other influences:
The geometry of the sky distribution has a significant impact on the displacement of the antenna.

Zero Baseline Simulations
This was used to demonstrate the impact of sky distribution for three different sites in the EUREF network.
\begin{itemize}
\item 2 sets of observations from one site are processed
\item bot sets of observations are identical
\item One set is corrected for with a relative PCV
\item The other set is corrected with an absolute PCV
\end{itemize} 
This should give us a baseline that only shows the impact of the transition from relative PCV to absolute PCV for this site.
The simulation was carried out for 6 different antenna types.
The results showed there was less of a vertical / horizontal effect when the sky distirbution was good (no overhead hole)
The largest effect was with the radome TCWD at NYA1 which experience a 20mm change in height.

The zero-dffierence approach only shows the influence of different antenna calibrations in local networks which do not exceed a couple of km.

In regional network solutions, a troposphere correction will have to be estimated, which can differe significantly between sites with a spacing of a few hundred to few thousand km.
Uncorrected PCVs lead to errors in the troposphere parameter estimation and will amplify the change in height component drastically.
Satellite geometry is a function of the two sites which form the baseline.
Baselines are computed between the LROC , where an AOAD antenna is installed and the 3 previous stated sites.
The results show that the differences between the abs/rel calibration models were much larger than the Zero-difference simulation.
Each antenna type has its own charcteristics/ displacement effect.
The smallest change appears in the LEIAT504 antenna. this doesn;t mean it is a better antenna, just that the absolute and relative calibrations models agree to a better extent.
Changes in height can be very large (up to 30mm).
The author believes that the stations he checked were antennas which built in a very similair manner (choke ring, same element) but the DOMES were completely different.
The radome will change the reception behaviour of the antenna, which can cause a change in position.

Plots are then shown which display the difference between calibration solutions for antennas without a radom and a specific radome installed.
This is achieved by substracting the elevation depenednet PCV on one antenna with a DOME and with the same antenna without a DOME.
All of the DOMES examined do not exhibit a perfect behaviour (however author doesn;t specify what frequency he looks at)
The perfect dome for positioning would have the same correction value for the complete hemisphere.
The signal delay caused by a perfect dome would then be attributed to a clock error of the receiver, and would not contirbute to a pos error.
This may have a signifiicant impact on Time Transfer Techniques..

When combining Geodetic techniques, only coordinates which reflect the physics can be used in combination with other geodetic techniques.
Using Absolute Antenna Calibrations brings us closer to a set of coords that reflect the underlying physics.

\subsection{Questions to follow up...}
\begin{enumerate}
\item What are the effects of using the NGS relative calibrations converted to absolute
\item is the difference mainly due to NF or technique based as well?
\item Why is the scale of the reference fram held fixed from the IGB00 solution?
\item Won't this distort the errors that absolute antenna models are meant to be correcting?
\item Can NF suscpetibility of an antenna be estimated by looking at the difference in calibration results is chamber vs robot, etc...?
\item Need a program which will:
	\begin{enumerate}
	\item Plot elevation depenedent errors from the Abs cal file
	\item Plot azimuth and elevation error
	\item Show the difference between two antenna types
	\item Need to be careful of the 'copied' type correction models.
	\end{enumerate}
\item Redo YAR2 survey in GAMIT
\item What effect will a Referance Frame stations without a correct antenna model have on a network solution? ie YAR2
\item Do a zero difference with and without the relative cal model?
\item Look at very local network, ie YAR2, YAR3, YARR
\item Look at small regional network is AUSPOS
\item Look at large regional network like APREF
\end{enumerate}


% ==========================================================================================================================================
\section{Absolute Calibration of GPS antennas: laboratory results and comparison with field robot techniques.}
\citep{gorresAnechoic2006}

Summary of paper:
\begin{itemize}
\item A critical assessment of the accuracy of GPS antenna calibration is mostly effectively carried out by comparing different methodologies
\item Compares Geo++ robot with chamber and relative calibration techniques
\item Validated the chamber and Geo++ calibrations to agree within 1mm of each other
\item confirm presence of significant variation between different antenna types
\end{itemize}

The Properties of the incoming signal are distorted by the electriv field projected by the antenna.
One driver for antenna corrections is th eneed to use mixed antenna types in RTK networks.
An accurate phase center variation model fosters a shorter time to fix the ambiguities.

GPS antennas were calibrated in a large anechoic chamber belonging to the German Armed forces.
Validation of antenna calibration is only possible by comparing results of individual antennas by different organisations and by analysing the results from different methodologies.
Previous comparisons were only done on an antenna type basis, if there is a difference it makes it hard to determine what the cause might have been.


Antenna Calibration Model
The adopted antenna calibration model is based on the well-known antenna phase center variation correction equation. where the total phase center correction in the direction of the satellite consists of the absolute mean antenna phase center offsets (PCO) with respect to the antenna reference point (ARP), plus the elevation and azi dependent phase center variations (PCVs) given in degrees.

dr(alpha,beta) = a . r0 + lambda . dtheta(alpha,beta) 1(a)

with PCO a(lambda) = (ax,ay,az) where lambda is the wavelength of carrier phase.

The estimation of this model is carried out in two steps.
First step contains the estimation of the 3 components of the mean PCO wrt to the ARP.

\[
\sum \delta\theta(\alpha,\beta)^2 = Min - 1(b)
\]

ARP is conventionally defined as the intersection of the veritcal antenna axis of symmetry with the bottom of the antenna. For anechoic chamber solutions th ephase pattern is obtained directly as residual from the adjustment of 1(b)

In a second step the pattern can be modelled using harmonic functions in order to estimate the measurement noise and the smoothness or quality of th ephase pattern.

For the first setup they used the 2D functional model for the PCVs
\[
\delta r (\beta) = \sum (ak cos \beta + bk sin \beta) k = 0, .. 3 or 5
\]
from the difference between the modeled and measured data the RMS scatter is calculated to provide a measure of the scatter of the phase pattern.


For several years relative phase patterns have been used by the GPS community that can be estimated from GPS measurement on a short baseline. According to the IGS standards the relative phase PCC are based on the absolute values for the PCO of the reference antenna (DMT) estimated from chamber measurements and upon the arbitrary assumption that the PCVs of the ref.antenna are zero.

Relative calibration in the field do not permit a homogenous distribution of observations with regards to the antenna hemisphere.
In addition the contain site specific effects. (but some would argue that this is present in todays absolute calibrations)

Absolute calibration corrections for the recevier antenna can be obtained from 2 independent techniques
(1) measurements in an anechoic chamber
(2) field measurements on a short baseline using a robot mount

The authors were unable to achieve a complet hemisphere coverage due to a shortage of time in the chamber.

German benchmark test 2002
Several institutes carried out a benchmark test in order to study the performance of different calibration methods (Rothacher et al. 2002).
They selected five antennas (5 different types from 3 different manufacturers)
Two institutes used an absolute field technique with robot
Three others used standard field calibration in relative mode with a reference antenna
It was not possible to carry out chamber tests at the time of this test.
Official comparison was presented by Schmid et al 2002

The robot results had a good comparison at the 1mm level.
The relative field technique displayed variation of 2mm at L1 and 4mm at L2.
The relative technique also showed a problem at the highest and lowest elevations caused by the small number of obs at zenith as well as systematic effects at the horizon.
If the results are expressed in RMS differences over all elevations, it can be seen that the absolute PCVs derived from the relative cals are a factor of two worse than those obtained from absolute techniques.

Anechoic chamber antenna calibration

They used a very large chamber 41m (l) x 16 (w) x 14 (h)
They borrowed a transmitting antenna from the MAX Planck Institute that can produce RHCP radiation
The absorbing material of the chamber was designed for frequencies from 0.5 Ghz upwards.
The distance between the receive and transmitter was 18m
Transmitted and received signals were compared in a network analyzer.
Recordings of the phase delay and Amplitude were performed for both frequencies L1 and L2.
The location of the center of rotation of the test antenna had to be determined with high accuracy wr to a physical point on the test antenna.

Results
measurements of the elevation dependent PCVs show very smooth patterns.
no fitting functions were applied to the measurements.
The resolution of the phase measurements was about 0.1mm
For L1 results 2 types of pattern may be distinguished
\begin{enumerate}
\item patterns with two maxima common to most 'new' antennas
\item patterns with three maxima for old Trimble antennas
\end{enumerate}

The smallest azimuth variations where for the LEIAT504 and Trimble Geodetic
Some antennas show large azimuth variations with oscillations with 180 period AT303 (L2) and Trimble Geodetic (L1)
Two groups of antenna can be clearly distinguished again:
\begin{enumerate}
\item older antennas with a large scattering
\item newer models with a distinct smooth pattern
\end{enumerate}

When they compared there results with the robot there was a small azimuth phase shift.
They thought this was due to the uncertainty of the antennas orientation in the chamber set up.

When they compared with the robot they had to consider 
\begin{enumerate}
\item disagreement in offsets
\item agreement in shap of the pattern
\end{enumerate}

They made an additional adjustment and estimated resdiual offset differences (height component) between the phase pattern of chamber and robot.
For asymmetric patterns the difference increased to 4-5 mm
Where near 0 for AT303 and Zephyr

The difference can be explained by the improvised measurement of the mechanical center of rotation relative to the ARP.
They only measured 1 meridian and 1 parallel circle and not the complete hemispherical pattern.
There exists variation among the patterns of individual antenna of the same type.(Wubbena 2003)

When the error in angle of rotations is considered for the chamber tests, the quality of the PCV fit with the robotic solutions improves dramitically.

Chamber measurements have the potential to study the characteristics of the PCV patterns in greater detail and at a higher level of precision as no fitting function needs to be applied.

Questions to follow up:
\begin{enumerate}
\item What are the concerns with an artificial signal in a chamber?? Just code related, or phase as well?
\item What about using a pseudolite in a chamber for calibrations, will this make it a more realistic signal?
\item would we get an accurate group delay, etc??
\item Could we use a pseudolite mounted on a robot in the field for absolute calibrations?
\item What is the NMI chamber foam absorbing charcaterisitcs designed for? what frequency range? What are the dimensions, etc?
\item If we can veryify that converting relative to absolute calibration is valid for Regional processing (YAR2) 
	\begin{itemize}
	\item Then we can compute a lon time series of antenna calibration PCV/PCC for YAR2
	\item Then use this antenna model over a long time series to determine coords.
	\item Could try a solution only using YAR3 (already has a absolute calibration) to compare a solution with YAR2
	\item Then could try a solution only using YARR to compare a solution with YAR2, YAR3
	\end{itemize}
\end{enumerate}

% ==========================================================================================================================================
\section{Tests of Phase Center Variations of various GPS antennas and some results}
\citep{schmitzResultsAbsolute2002}

The errors in PCV affect long-term static GPS differently than RTK GPS.

Mixed rovers for survey applications may be more important to have an absolute calibration.
PCVS were determined and analyzed in laboratory experiments from the very start of the development of GPS. 
Sims(1985) reporterd absolute values of elevation and azimuth dependent PCVS fo rthe antenna of the first civillian T14100 GPS receiver.

In practise the same types of antennas were used and antennas were orientated in the same direction, so that over short distances the PCV effect was not signficiant.
At this time is was sufficient to measure the height of the antenna to the same reference point.
It was not possible to distinguish PCV effects on static measurements over long periods at short to medium ranges from other effects.

When it became common practise to use different types of antennas and measurements were being made over long distances, the satellites had significantly different viewing anglesand the effects of different PCVS was becoming apparent.

Absolute calibration from anechoic chambers and their analysis are reported by Schpler (1994), Schupler et al. (1995) and Rohan et al (1996).
Obtaining the absolute PCV was a relatively costly lab calibration and did not give satisfactory results in practise.
(Scale bias issue?)

This led to the introduction of field procedures for antenna calibrations. 
They were able to estimate elevation dependent PCVs in relation to a reference antenna.
The PCVS of the reference antenna were assumed to be 0, and offsets were fixed to a predetermined value (Rothacher and Mader 1996).
The absolute PCV component of the reference antenna, with a variation of 28mm for the ionosphere-free linear combination (L0), is not corrected and introduces a systematic error (Wubbena et al 2000b).

Multipath is a critical factor in relative calibration procedures and there have been several attempts to minimise its effect by site selection and averaging over time.
The PCVS at this time were commonly applied from estimatations from an uncomplicated field procedure (Mader 1999).
Freqeuntly only manufacturers data on the mean L1 offset, sometimes the L2 value were used in the computations.
Alternatively offsets were determined in seperate rotation test measurements or taken from published calibration results. 
PCV corrections were generally not taken into account.

However due to the increasing use of different antenna types, and with the increase in number of permanent reference stations and RTK networks the need for full PCV corrections became apparent.

Initially comparison of relative and absolute PCVSan effect emerges in global networks which represent a scale error of 15 ppb as a first approximation (Rothacher et al. 1995, Menge et al. 1998).
on the basis of results of measurement techniques other than GPS, absolute PCVs were first thought to be the source of this error. The cost of determining absolute PCVs in a lab inhibited further investigations.
Experiments have now shown that absolute PCVs can be excluded as the cause of the so-called scale errors in global networks. 
New investigations into other sources of error, such as the satellite antenna and troposphere were required.

The first field calibration of a GPS satellite antenna PCV was conducted in 2001 (Mader and Czopek 2001).

At first it was thougt elevation-dependent PCV was the largest error component, and azimuth PCVs were thought to be an order of magnitude smaller.

Through an increase in resolution, accuraccy and reliability resulting from absolute field calibrations shows that some antennas have significant azimuth variations.
This has proved to be particulraly important for RTK applications.

Absolute Calibration in the field
The effects of the station dependent error components (PCVs amd multipath effects) are at first indistinguishable.
It is essential for absolute calibration the seperation and determination or elimination of the various individual source of error.
GEO++ use an undifference approach.

The first procedure relied on sidereal day differences (Wubbena 1996,1997).
Further developments led to the operationally available uses of short-term observation differences for the purpose of MP elimination.
Sidereal approach the MP error term is eliminated or removed on the condition indentical MP existson the replications of the sat constell.
The PCV difference between static obs on a ref day and the obs with rotated or tilted orientation on a 2nd day are used as input for the PCV determination using spherical harminc analysis.
Any influence of a reference antenna are removed as no changes of orientation occur.
All other paramters in the GPS adjustment are eliminated or estimated. Details are given in Wubben (1997), Seber(1998), Menge(1998).

The undifference observation equations forms the basis for the Real-time calibration process.
In addition to standard parameters of GPS adjustment for short distances, the high correlation of MP effects between successive epochs is used to estimate MP as a stochastic processes.
The rapid changes of orientation of the robot allow the seperation of PCV and MP effects. In addition comprehensive calibration of the robot itself is required through theodolite observations to enable antenna positions to be determined to about 0.2 to 0.3mm.

Accuracy requirements for PCV corrections

The standard deviation of the L0 PCV is about 3 times of L1 and L2 PCVs. Generally we will need to be at: 
\begin{itemize}
\item 1 to 2mm for L1 and L2 offsets
\item 1 to 2mm for L1 and L2 PCVs
\end{itemize}

This will give a standard deviation of 4 to 7mm for L0.

The effect on GOS position accuracy is defined in terms of standard deviation of position Sp and the standard deviation of phase measurements Sr, together with a measure of the sat geometry the PDOP as follows

sP = sR x PDOP

With a PDOP value of 1 to 3, uncertainities in the PCV effects give rise to possitional uncertainites in the range of 4 to 21 mm.
This is not good enough for high precision applications.
Therefore the best possible accuracy for the combination of offset and PCV is required for PCV determination.


Effects of UHF aerials in the vicinity of the GPS antenna

Additional aerials in the immediate vicinity of the GPS antenna elements fundamentally alters their electromagnetic reception properites.
An antenna was calibrated with and without an attached UHF aerial.
The UHF aerial was passive, ie not transmitting.
The difference in PCV reached +/-12mm
The UHF aerial in the vicinity of the GPS antenna is not small enough to be considered negligible.

PCVs of integrated antenna and receiver system
The antenna is placed above the receivers electronics in the same housing.
Get large azimuth variations (which could be due to the shape of the housing)
Can get large azimuth variations at L0 of +/15mm for the Zeiss.
However the integrated Trimble TRM4800 system only has very small L0 PCVs of +/- 2mm.

So in principal small PCVs and virtually symmetrical PCV charcateristics are possible even with an integrated system, given sufficient shielding combined with a suitable housing. However it cannot be generally assumed that every model of integrated antenna exhibits this.


Effect of antenna dome on PCV

In general an antenna with a dome is regarded as a different type of antenna with its own characteristics.
However the effect is often underestimated and ignored.
A constant difference in PCV does not appear to be important, if the combination can be calibrated.
However any difference which gives rise to a region of steep gradients will adversely affect the quality of GPS observations.

The Trimble TCWD dome is fixed with a metal band onto a seperate ground plane beneath the antenna. 
The large additional ground plane changes the absolute PCV for L0 over a range of 16mm. 
The changes are mainly at low elevation but do extend above the usual elevation masks.

Several other types of dome without additional GP such as the Ashtech SNOW and LEIC have smaller effects +/- 1 or 2mm. 
The conical dome LEIC only introduces small effects below 30 at around the 2mm level.

Significant differences in PCV are also caused by the design of the dome (in terms of material, etc) and how close it is to the receiver elements of the antenna.

The LEIS and SCIS dome have different affects on the AT504:
have a similar spherical shape, but made of different material
SCIS uses steel screws and is made of a harder plastic, its diameter is about 3cm greater than the antenna so the complete hemisphere cover with vertical sides.
The LEIS screws directly to antenna forming a hemispherical cover with vertical sides.

The PCVs of the same antenna with different domes, have clear differences of about 6mm incrreasing to 12mm between 25 elevation and horizontal.
When the results are compared to the antenna only calibration, the LEIS has less of an impact on the original PCV,
SCIS clearly changes between 90 and 60 with gradient exceeding 1mm/degree.


Effect of a Ground Plane on PCVS

The TRM22020 has a 48cm ground plane. Without a GP we get large azimuth variations which are correlated with the corners and sides of the angular antenna housing.
With the GP the reception behaviour changes radically, there is no comparable azimuth variation in PCV, but there are large elevation dependent PCV gradients of up to 1mm/degree at high elevations. Any variation in the GP diameter, thickness and material will change the antenna characteristics.

PCVs of rover antennas
Are charcaterized by their small diameter making them lighter and more manageable.
The small ground plane diameter is presumbably the reason for their PCVs often having a very small range.
They can come close to representing an absolutely calibrated point-form antenna.

The NovAtel 600 according to the manufacturer exhbits no positional offsets, with identical heights for L1 and L2.
They claim no elevation or azimuth dependency. 
The calibration of the NOV600 shows an increased noise level on the observations in relation to comparable antennas.
Position offsets are in the order of 1 to 2mm.
Difference between L1 and L2 height offsets are about 3.7 mm.
The L0 PCV of the NOV600 above 10 exhibits variations of about +/- 1 to 2.5mm and no significant azimuth PCV.

The LEIAT502 also shows small PCV values (although some large at the horizon)
JPSLEGANT is also small azimuth PCV of +/- 1.5 mm, but has clearly a larger and predominantly elevation dependent PCV component.

For high precision work and especially for precise real-time applications individual calibrations are imperative.
The factors affecting PCV charcteristics are too complicated for them to be described in general terms.
Significant alterations are caused in the neighbourhood of the antenna by:
\begin{enumerate}
\item UHF aerials
\item shape of the antenna housing
\item details of the construction of the dome
\item presence of a ground plane
\end{enumerate}

\subsection{Questions to follow up:}
\begin{enumerate}
\item a table showing PCV estimation vs GPS error budget by components, is troposphere, ionosphere, etc and time
\item how do you calculate the azimuth only dependency and still show a 3d plot??
\item what might the steep gradient impacts of the SCIS dome have on positioning results?
\end{enumerate}

% ==========================================================================================================================================
\section{Challenges for GNSS-based high precision positioning - some Geodetic aspects}
\citep{schonChallenges2007}

MP, signal difraction and incomplete stochastic models for GNSS limit the attainable accuracy.

Site-dependent errors like MP and diffraction as well as the incompleteness of the stochastic model for GNSS obs are still open issues.
This is particularly important for RT and rapid static as these disturbing effects can distort the estimated position on the mm to cm level.
The bias degrades the speed accuracy and reliability of ambiguity resolution and position determination.

Mulitpath Propagation
The signal reaches the antenna via more than one path.
The antenna will capture the direct signal (LOS) superimposed by one or more of its reflections from objects in the vicinity and from the ground.
The MP error will appear in the basic observables such as carrier, code as well as signal to noise.
Typically it appears as periodical oscillations with periods between 15-30 minutes.

For one reflector:

\[ 
\delta\phi = arctan( \frac{sin(\delta\phi)}{\alpha + cos(\delta\phi)} ) 
\text{where} \alpha, 0 <= \alpha <= 1 \text{denotes the amplitude attenuation}
\alpha = 0 \text{no reflection}
\alpha = 1 \text{identical strength of direct signal}
\]

The phase shift between direct and reflected signal is given by:

\[
\delta\phi = \frac{2 \pi d}{\lambda}
\text{where d = excess path length of the reflected signal with respect to the direct signal}
\]

The amplitude of these effects are given by

\[
\delta\alpha = \sqrt{ 1 + 2 \alpha cos (\delta\phi) + \alpha ^2}
\]

The magnitude of the effects reachthe dm to m level for code, and several cm for phase.
The maxima for the single reflection scenarios are:
\begin{itemize}
\item 4.7 cm for L1
\item 6.1 cm for L2
\end{itemize}

Forming different linear combination of the original phase observation, the maximum mulitpath error canincrease by factors of between 2 to 9. 
The patterns and magnitudes of MP effects depend on:
\begin{enumerate}
\item The ratio of the delay, phase and amplitude of the reflected signal with respect to the direct signal.
\item The number, size material, roughness and reflectance of reflectors.
\item satellite-reflector-antenna geometry and spatio-temporal changes
\item Hardware and software used during the GNSS measurements and their analysis.
\end{enumerate}

Multipath mitigation strategies:
Can fall broadly into three different categories:
\begin{enumerate}
\item Pre-receiver approaches
\item Dedicated tracking techniques
\item Post receiver signal processing
\end{enumerate}

Pre-receiver techniques usual involve:
\begin{enumerate}
\item carefule site selection and antenna location
\item specially designed antennas providing a properly shaped gain pattern, that has a sharp roll-off near the horizon and rejection of LHCP signals.
\item choke ring antena
\item antenna arrays with multiply closely-spaced antennas are used to estimate and eliminate multipath in RT.
\end{enumerate}

Dedicated Receiver based tracking techniques rely on the optimal design of the correlators, and the shape of the disciminator function implemented in the delay lock loop (DLL) and phase lock loop (PLL). Examples include:
\begin{enumerate}
\item Narrow Correlator
\item MP Elimination Tecnique
\item MP Estimate Delay Lock Loop
\item Strobe and Edge Correlator
\item Enhanced Strobe and Edge correlator
\item Maximum Likelihood Estimate
\item Vision Correlator
\end{enumerate}

It should be noted that a separation of diret and indirect signals is only possible for direct multipath delays > 10m.
Strong signals from near the antenna cannot be eliminated or mitigated using the above techniques.

Post Receiver Signal Processing:

In geodesy the focus is mainly on post-receiver signal processing techniques applied to the estimated coordinate time series. 
Some static processing techniques use the sidereal difference of GPS observables of consecutive days.
This assumes a strict temporal and spatial repeatability giving identical MP effects.
However the actual revolution of GPS satellites drfits over time and can deviate up to 8s from the nominal period.
In fact each satellite has its own period.
The ground track of GPS satellites spatially varies due to nodal precession.
The nominal variations of the argument of the ascending node is specified within +/- 2 .
Maneuvers are flown by the GPS Control Segment to reposition satellites on its nominal orbit.
The maximal variation translate into sepratiions of the ground track by several dm after 1 day. So different objects will get scanned and this may reflect in a different manner, yeilding a different MP pattern.
However sidereal filtering does lead to significant improvement of the coord time series.
But this may be filtering other signals at the same time, suchas a atmospheric and geophysical signals that maybe of interest.

The obtained coordinate time series can contain a superposition of the mapped multipath effects of all observations. So a detailed study of the different factors creating the MP effect is difficult on the basis of the time series alone.

There is also a signal-to-noise aproach, which allows for the reduction of low frequency MP to the obs noise level. Its implmentation suffers from different assumptions needed especially about the antenna gain pattern and the MP frequencies.

Some authors have propsed a diagnostic approach for regional GPS networks.
Step 1 Detection and localisation of MP effects by analysing the DD ionsphere free residuals.
Daily computation of MP correction maps for the undifference original observations. 
Application of this technique could produce significant reduction of MP errors in common linear combinations, but not for the original L1 and L2 signals.

A prototype antenna was developed to act as a Multipath calibration system, by using a steerable parabolic receiving antenna.

Despite these numerous investigations on different techniques, the global correction strategy seems to be still missing.


Station Calibration

An alternative approach tacken by GEO++ and University of Hannover consists in the MP calibration of permanent GNSS stations.
It is based on the field calibration strategy that has been successfully applied for the determination of the absolute antenna pcv with a precise robot.
For the purpose of MP calibration a 2nd GPS equipment is used and its antenna is placed on a robot which is mounted near the reference station to be calibrated (< 5m).
MP effects are very site dependent, ie a small change of the antenna location leads to a completely different signal pattern.
The basic idea of the calibration process is to decorrelate the MP on the first station due to the pseudo random motion of the robot.
The short baseline seperating both stations will have a nearly identical obsrvation conditions so atmospheric and orbit erros cancel out.
If the precise PCV of both antennas are known, the MP pattern of the reference station can be estimated.
This is only true if the MP signal can be seperated from the receiver clock errors.
So the technique uses an external oscillator to steer both clocks.
A Satellite by satellite coefficient of Tschebycheff polynomials are determined for each carrier signal to encounter the MP effect.
At this point the adjustment can be constrained using points of intersection between different sat orbits.
The plynomial coeeficients should represent the MP behaviour at the station to be calibrated.
This alows the data to be corrected for future observations.

Detailed studies have been carried out to analyse the temporal stability and variablity of the obtained polynomial coefficients.
During the analysis at different sites and with different equipemnts, it turned out that a conceptual separation of the MP effects into near field and far field can contribute to a better understanding and modelling.

Near and Far Field Multipath:

GNSS receiving antennas are generally deirectly mounted on larger structures (concrete pillars or pylons). These mounts add additional reflecting surfaces that can create additinal scattering of the observations.
In addition the mount is electromagnetically coupled to the antenna. Therefore the inital PCV pattern of the antenna will be modified.
Conceptually it seems to be beneficial to seperate the MP effects into NF and FF.
NF effects shaw rather long periodic oscillations with a non-zero mean value.
So these effects do not average out when the observation span is increased (eg 48 hours), and will lead to a bias in the coordinates.

GEO++ propose to carry out a NF calibration including the antenna mount.
Different mounts can yield deviations of several mm in the PCV.

The major challenge in geodesy is to guarantee the long-term stability (< few mm) of the coord time series for the realisation of a reference frame and to avoid jumps in the time series.
This is especially true during the necessary update and upgrade of the station equipment to be compatable with modernized GNSS.

Cooridnate time series can have jumps/discontinuities of up to some cm even if the antennas are absolutely calibrated.

Today an operation procedure is missing to select the optimal placements of reference station networks.
A software receiver may be a promising new tool to learn about the original MP patterns so that they are not biased by the tracking loops.
In addition the original carrier phase measurements are stored and can be reprocessed by different analysis strategies allowing to optimise the processing for typical MP patterns.

Signal Diffraction:
This occurs when the GNSS signal wave fromt arc is partially obstructed but not totally blocked.
Some of the signal is received becasue it is diffracted around the object:
\begin{itemize}
\item passing through foliage
\item around iron objects
\item construction in LOS to sat (ie suspension bridge)
\end{itemize}

Signal diffraction is always associated with changes in the C/No. In general the diffracted signal has a lower C/No than a direct signal.

Diffraction Mitigation Techniques:

The exact delay and ecsee of path caused by difraction is dificult to assess, diffraction effect are not corrected.
However the impact of diffracted obervations can be down weighted so their impact on the estimatin process can be minimized.
The magnitude of the C/N0 density ratio can be directly used for down-weighting.
Low C/N leads to large variances (so these obs are given a smaller weight)
It has been propsed to use the Sigma-e for this task.
The model can be improved further if C/NO has reference values for the equipment used.
The reference values should refer to diffraction-free LOS measurement from all azimuth and at all elevation angles.
Assuming azimuthal symmetry a common, elevation dependent template function can be derived, this is realised in the SIGMA-Delta model.
Not only the actual C/No value of an observation but also the magnitude of its deviation from the template can be used to determine the observational variance.

% ==========================================================================================================================================
\section{CORS local-site finger-printing using undifference least-squares GNSS phase residuals \citep{huisman2009}}

CORS local-site dependent effect, such as multipath can be seperated from atmospheric delays by stacking undifference least squares residuals of a network from a period of 2 to 3 weeks.
Analysing the undifference phase residuals at a single site can reveal an azimuth and elevation dependency of the residuals.
Different time periods are used to give insight into the repeatability and stability of the estimated finger-prints.
Introducing the local-site MP and antenna effects, results in a more consitent empirical standard deviation fo rthe undifferenced residuals.

Despite numerous models and mathematical cancellation techniques the results still contain GNSS systematic errors.
If we assume that:
\begin{enumerate} 
\item satellite orbits, 
\item satellite-clock errors, 
\item mean antenna phase centre variations, 
\item mean atmospheric delays and 
\item carrier phase ambiguities 
\end{enumerate}
were correctly modelled in the data processing, then the undifference residuals will be dominated the receiver carrier phase multipath, un-modelled antenna phase centre variations, and unmodelled atmospheric delays. (Alber and Van der and Gundlich) 
If we assume the unmodelled atmospheric delays vary from day to day and are zero on average they will cancel out when undifference residuals of multiple days stacked.
Stacking the undifference phase residuals of multiple days will therefore give the systematic errors in the residuals that mainly consist of receiver multipath and unmodelled antenna phase centre variations.
Rocken used this approach to analyse and explain systematic single epoch positining errors at CORS network sites.
Another approach is to determine the MP effects and then use this to improve the estimation of slant wet delays (Braun2001 De Haan 2003, Van der Marel2006 and Marcias2007)

The receiver multipath and un-modelled antenna phase centre variations can be visualized by a multipath map (Braun1999) where the values of systematic phase residuals are plotted in a spherical map against azimuth and elevation.

Tried a small network around perth, but it was too sensitive to missing data which gave jumps in the coordinate time series. So they decieded to use the larger network.
Processed in Bernese.

The double difference residuals are residuals of the ionosphere free linear combination of the L1 and L2 GPS observables.
The method of obtaining undifference residuals from the DD residuals was introduced by Alber2000. 
The method is based on two assumptions:
\begin{enumerate}
\item The transformation from undifferenced observations to single difference observations between receivers the satellite clock errors cancel out.
    \begin{itemize}
    \item Since the single epoch satellite clock error is the same for all observations to that satellite the weighted mean of the undifference residuals should be zero.
    \end{itemize}
\item When doubled differences are formed receiver clock errors cancel out.
    \begin{itemize}
    \item Since the single epoch receiver error is the same for all observations at a receiver the weighted mean of the single difference residuals should be zero.
    \end{itemize}
\end{enumerate}
By using these zero mean conditions, DD CORS network-processed phase residuals can be transformed to undifferenced phase residuals of the ionosphere free linear combination.
Iwabuchi2004 showed that applying the zero-mean condition on the dd residuals from the bernese GPS software gives similair results from PPP with the GIPSY software.

How the Multipath Maps are generated:
The undifferenced phase residuals depend on the satellite to receiver geometry ad thus depend on azimuth and elevation of the observables from a satellite.
To project the residuals in a multipath map the residuals are firts gridded as a function of azimuth \[\alpha\] and elevation \[\varepsilon\] in a sinusoidal equal are projection.
A grid size \emph{s} is chosen in the sinusoidal equal area projection so that each grid cell describes an equal area on the sky.
Except fo the azimuth \[\alpha_0\] the azimuth in the sinusoidal projection is a curved line, the largest curve is in azimuth \[\alpha_0 + 180\].
For multipath maps in the Southern Hemisphere \[\alpha_0 = 180\] is chosen as this results in the largest distorion in the southern direction, where no data is available due the GPS constellation sky hole.
The multipath value m of a grid point in multipath map a is now calculated from the mean of all the undifference residuals \emph{r} that are inside the area \emph{S} that surrounds that grid point. Each undifferenced residual can only be in one area \emph{S}.

\[
S_{a,c} = \{(x,y)
\]

The method described above to combine all the observations inside the area \emph{S} is referred to as stacking by Alber (2000). 
The equal area projection is the same as has been used by Van der Marel \& Gundlich (2006), but modified for the Southern Hemisphere.
The GPS constellation will not cover the full field of view at a station, so not all areas \emph{S} will contain one or more observations. Therefore not all grid points \[(\alpha,\varepsilon)\] will be assigned a value in the multipath map.

Multipath maps can show blockages caused by trees, ie variations in azimuth, and areas influence by multipath.

\subsection{ Repeatability of Multipath Maps}

Ideally the multipath map only contains systematic errors due to MP. 
If this is the case then the difference between two multipath maps is expected to be zero at all the grid points. 
However we do expect that the MP maps for each day wil contain a systematic error due to un-modelled atmospheric effects, in this case the difference between multipath maps will not be zero.
The residual \emph{e} at a grid point \[(\alpha,\varepsilon)\] between the multipath maps of a single day \emph{i} and the multipath maps of \emph{j} days is:
\[
e_i(\alpha,\varepsilon)=m_i^1(\alpha,\varepsilon - m^j(\alpha,\varepsilon))
\]
After stacking the residuals of a certain number of days the un-modelled atmospheric delays will have averaged out and only remaining systematic errors (MP and un-modelled PCV) remain.
The empirical standard deviation for a grid point \[(\alpha,\varepsilon)\] will not change significantly from that day as the systematic errors are expected to be constant with a certain standard deviation.

\[
    \hat{\sigma_{e^j}(\alpha,\varepsilon)} = 
\]



\subsection{ Follow Up:}
\begin{itemize}
\item Look at different methdologies of interpolating the residual map, are there any numerical effects that should be considered?
\item What time period should be used
    \begin{itemize}
    \item Antenna changes only
    \item two - three weeks 
    \item 3-4 months
    \item 6-8, etc..
    \item What is the minimum time to generate a model with a high level of certainty(pre chi, post chi) required to generate a model, etc..
    \item will the different time period averaging have an impact on the systematic errors that are be acocunted for by the model?
    \end{itemize}
\item What are the effects of using different processing methodolgies, small, regional, global networks, change of reference stations, etc..

\end{itemize}

% ==========================================================================================================================================
\section {Obtaining single path phase delays from GPS double differences}
\citep{alber2000}
The paper describes a methodology of obtaining single path delays from GPS double differences.
The resulting ZDs can be used for remote sensing of the atmospheric water vapor.
In-situ GPS antenna phase center and multipath effects are mapped in ZD residuals for a specific site and network.
ZDs derived from GPS network data show promise for real time sensing of water vapor for use in meteorological modeling and forecasting.

They claim using this approach that the precipitable water vapor (PW) above the GPS antenna can be estimated with better than 2 mm accuracy.

High accuracy GPS applications commonly use double difference to cancel satellite and receiver clock errors and to help determine integer values for Carrier phase ambiguities.
However DD include observations along 4 different paths (from 2 observation satellites to 2 satellites).
The DD observations are more difficult to interpret than single path delays.

Double difference can be expressed as the difference of two single differences. 
DD cancels common mode errors such as satellite and receiver clock errors.
Depending on the network size, and equipment used, errors from satellite orbits, atmospheric delays, site coordinates antenna pcv and MP effects may or may not cancel out.

DD can be converted to SD if an additional independent constraint on at least one of the SD is made.

In the inversion from DD to SD and SD to Zero Diff any error in the zero-mean assumption is divided equally over all stations in the network.
The ZDs are relative to the ensemble mean of the network.
This implies that large networks can be used woth careful modeling to minimize biases. 
Station coordinates should be held fixed to knwon long-term averages to avoid correlation with slant delays.

For tropospheric delays the zero mean assumption implies that the residual delay in the direction of one GPS satellite at each epoch, averaged over the entire GPS network is equal to zero.
For a GPS network that is distributed over a large area (~100km) this assumption is generally valid because the distribution of water vapor at the sites can be considered random and the mean zenith delay can be estimated and removed.
For a small network, if all stations observe a satellite through the same volume of atmosphere, the delay will have common mode elements that can not be resolved.

Residual phase variations that repeat with the periodicity of the GPS orbits can be mapped with the ZD method. Assuming that site coordinates, carrier phase ambiguities, GPS orbits, in-situ antenna phase center variations, and mean atmospheric delay are accurately modeled, the ZD resdiuals wil be dominated by the un-modelled atmospheric slant delay and MP effects.

Detailed multipath maps can be computed by stacking daily GPS residuals. 
Multipath effects can be corrected by substracting the stacked map from the residuals.
The stacking of residuals allows for the separation of multipath and unmodelled atmospheric slant delay.

% ==========================================================================================================================================
\section {Development of Models for Use of Slant Delays}
\citep{marel2006}

Slant delays are highly sensitive to site dependent multipath effects, which can result in errors of up to 3cm.
A strategy is needed to attempt to mitigate the effect of multipath.

\subsection{Line-of-sight residuals}
In order to compute the slant delays and correct for multipath line-of-sight residuals are required.
In the case of DD processing , the DD residuals have to be converted into undifferenced residuals.

The ZD residuals contain a lot of information that needs to be analyzed in a systematic way.
\begin{enumerate}
\item measurement noise and outliers,
\item site-multipath (different for all stations),
\item antenna phase center variations, depending on azimuth, elevation, antenna type,
\item orbit errors,
\item station position errors for fixed stations, and unmodelled station dynamics,
\item unmodelled tropospheric delays, the non-isotropic slant delay.
\end{enumerate}

In principle we can detect unmodelled systematic effects by collecting the residuals over many days and look for:
\begin{itemize}
\item common trends,
\item detect outliers in the data by analyzing unexpectedly large residuals
\item or by analyzing the residuals for multipath and unmodelled antenna PCV.
\end{itemize}

Once the systematic effects have been removed, the residuals are ready for computing slant delays.

\subsection{Multipath Mapping}
Multipath and antenna phase delays depend mainly on the azimuth and elevation at which a satellite is observed.
These errors change only slightly over time.
In order to analyze these errors, we will analyze the ZD residuals for each station as a function of elevation and azimuth.
The other errors that are still present in the ZD residuals, such as measurement noise, unmodelled atmospheric delays, do not repeat every day and therefor will cancel out over time.

% ==========================================================================================================================================

Antenna Design principles for IGS
\begin{itemize}
\item should there be a minimum set of standards of the antenna type installed at IGS stations?
\item should there be a set for reference frame stations?
\item what antenna charcateristics can be defined/specified? - smoothness of PCV?
\item minimum AR
\item susceptibility to Near-field multipath
\item LNA specs?
\end{itemize}

%\hline

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